Barrier Option - Valuation

Valuation

The valuation of barrier options can be tricky, because unlike other simpler options they are path-dependent – that is, the value of the option at any time depends not just on the underlying at that point, but also on the path taken by the underlying (since, if it has crossed the barrier, a barrier event has occurred). Although the classical Black–Scholes approach does not directly apply, several more complex methods can be used:

• The simplest way to value barrier options is to use a static replicating portfolio of vanilla options (which can be valued with Black–Scholes), chosen so as to mimic the value of the barrier at expiry and at selected discrete points in time along the barrier. This approach was pioneered by Peter Carr and gives closed form prices and replication strategies for all types of barrier options.
• Another approach is to study the law of the maximum (or minimum) of the underlying. This approach gives explicit (closed form) prices to barrier options.
• Yet another method is the partial differential equation (PDE) approach. The PDE satisfied by an out barrier options is the same one satisfied by a vanilla option under Black and Scholes assumptions, with extra boundary conditions demanding that the option become worthless when the underlying touches the barrier.
• When an exact formula is difficult to obtain, barrier options can be priced with the Monte Carlo option model. However, computing the Greeks (sensitivities) using this approach is numerically unstable.
• A faster approach is to use Finite difference methods for option pricing to diffuse the PDE backwards from the boundary condition (which is the terminal payoff at expiry, plus the condition that the value along the barrier is always 0 at any time). Both explicit finite-differencing methods and the Crank–Nicolson scheme have their advantages.